Up-and-down design

Dose-finding experiments have binary responses: each individual outcome can be described as one of two possible values, such as success vs. failure or toxic vs. non-toxic.

The goal of dose-finding experiments is to estimate the strength of treatment (i.e., the "dose") that would trigger the "1" response a pre-specified proportion of the time.

Dose-finding designs are sequential and response-adaptive: the dose at a given point in the experiment depends upon previous outcomes, rather than be fixed a priori.

Other UDDs break this symmetry in order to estimate percentiles other than the median, or are able to treat groups of subjects rather than one at a time.

[2][3][4] The 1950s and 1960s saw rapid diversification with UDDs targeting percentiles other than the median, and expanding into numerous applied fields.

[6][7] UDDs are still used extensively in the two applications for which they were originally developed: psychophysics where they are used to estimate sensory thresholds and are often known as fixed forced-choice staircase procedures,[8] and explosive sensitivity testing, where the median-targeting UDD is often known as the Bruceton test.

Each specific set of UDD rules enables the symbolic calculation of these probabilities, usually as a function of

Assuming that transition probabilities are fixed in time, depending only upon the current allocation and its outcome, i.e., upon

Usually, UDD dose-transition rules bring the dose down (or at least bar it from escalating) after positive responses, and vice versa.

[11] Since UDD random walks are regular Markov chains, they generate a stationary distribution of dose allocations,

This means, long-term visit frequencies to the various doses will approximate a steady state described by

According to Markov chain theory the starting-dose effect wears off rather quickly, at a geometric rate.

UDDs' central tendencies ensure that long-term, the most frequently visited dose (i.e., the mode of

Away from the mode, asymptotic visit frequencies decrease sharply, at a faster-than-geometric rate.

Even though a UDD experiment is still a random walk, long excursions away from the region of interest are very unlikely.

The original "simple" or "classical" UDD moves the dose up one level upon a negative response, and vice versa.

This UDD shifts the balance point, by adding the option of treating the next subject at the same dose rather than move only up or down.

makes this design identical to the classical UDD, and inverting the rules by imposing the coin toss upon positive rather than negative outcomes, produces above-median balance points.

Versions with two coins, one for each outcome, have also been published, but they do not seem to offer an advantage over the simpler single-coin BCD.

Some dose-finding experiments, such as phase I trials, require a waiting period of weeks before determining each individual outcome.

However, with cohorts viewed as single mathematical entities, these designs generate a first-order random walk having a tri-diagonal TPM as above.

For general group UDDs, the balance point can be calculated only numerically, by finding the dose

It was introduced by Wetherill in 1963,[14] and proliferated by him and colleagues shortly thereafter to psychophysics,[15] where it remains one of the standard methods to find sensory thresholds.

non-toxicities observed on consecutive data points, all at the current dose, while de-escalation only requires a single toxicity.

The internal state serves as a counter of the number of immediately recent consecutive non-toxicities observed at the current dose.

This description is closer to the physical dose-allocation process, because subjects at different internal states of the level

This approach antedates deeper understanding of UDDs' Markov properties, but its success in numerical evaluations relies upon the eventual sampling from

[18] In recent years, the limitations of averaging estimators have come to light, in particular the many sources of bias that are very difficult to mitigate.

However, the knowledge about averaging-estimator limitations has yet to disseminate outside the methodological literature and affect actual practice.

In 2002, Stylianou and Flournoy introduced an interpolated version of isotonic regression (IR) to estimate UDD targets and other dose-response data.

Simulated experiments from three different UDDs. 0 and 1 responses are marked by o and x, respectively. Top to bottom: the original "simple" UDD that targets the median, a Durham-Flournoy biased-coin UDD targeting approximately the 20.6% percentile, and a k-in-a-row / "transformed" UDD targeting the same percentile.
Examples of UDD stationary distributions with . Left: original ("classical") UDD, . Right: biased-coin targeting the 30th percentile,
Example for reversal-averaging estimation of a psychophysics experiment. Reversal points are circled, and the first reversal was excluded from the average. The design is a two-stage, with the second (and main) stage -in-a-row targeting the 70.7% percentile. The first stage (until the first reversal) uses the "classical" UDD, a commonly-employed scheme to speed up the arrival to the region of interest.